Non Linear Analysis by Finite Element Method of Rigid Slab Pavements in local Airports

Ali A. A. AlwashDepartment of Civil Engineering,University of Babylon

AbstractIn this research, the main factors influencing slabs analysis of local airport pavements were

studies. The finite element method is used for the analysis of single slab in pavement system. The type of element that adopted herein is the 8-noded (serendipity) element, with consider thick flat shell element has been considered. In other word, the membrane element is combination with bending element for result a shell element. The (ABAS-PROGRAM) is modified for have been ability for analysis the arbitrary quadrilateral shape with various boundary conditions. This program is used as nonlinear elastic on normal (linear and nonlinear) and tangential foundation. Additional, the layer approach is applied to complete the integration in z-direction after transferring it to isoparametric coordinate ξ in finding global stiffness matrix. The Cauchy elastic model is adopted for study the slab concrete pavement behavior.

The results presented within several applications. At start this application, the thickness slab pavement is taken in calculate the bending stresses and deflections under center of slab pavement. The subgrade reaction value is taken here for comparison the amount of deflection and bending stress for three moving type free, pin and fix boundary condition. Addition, the dimensions of single slab pavement are effect widely for free in side with pin and fix condition in other side about 28.57% for bending stresses and very large percent for deflection state.

الخالصة المطارات أرضية في الصلدة البالطات تحليل في المؤثرة العوامل أهم تناول تم البحث هذا في

تم الطريقة هذه المحددة. ضمن العناصر طريقة باستخدام الصلدة البالطات تحليل تم المحلية, حيث السميك, وهذه للسطح القشري العنصر نظرية تبني مع العقد الثماني نوع من المحدد العنصر استخدام تم ( قدABAS-PROGRAM) برنامج االنحناء. إن عناصر مع المحورية العناصر بتجميع تكوينها يتم النظرية حدية شروط تمتلك والتي أضالع أربعة تملك التي المختلفة األشكال تحليل على قادر وأصبح تطويرهمختلفة.

تربة على والمسندة الصلدة للسقوف الالخطي السلوك لتمثيل استخدامه يمكن البرنامج هذاالعمودي. لالتجاه بالنسبة والالخطي الخطي السلوك نظرية واعتماد والمماسي باالتجاهين, العمودي

( فيz) المحور باتجاه وذلك الكلية المصفوفة احتساب عملية عند إدخالها تم قد الطبقات طريقة إن لتوفير الدراسة هذه ضمن تبنيها تم قد الخطي للسلوك كووجي نظرية المحددة. إن العناصر طريقةالصلدة. السقوف بالنسبة الالخطي السلوك تمثيل إمكانية

اخذ تم التطبيقات هذه بداية التطبيقات. في من عدد إجراء عند عليها الحصول تم قد النتائج إن المركز. الحمل أسفل تحصل التي التشوه قيم وإيجاد االنحناء احهادات حسابات في الصلد البالطة سمك

والتشوه االجهادات قيم على ملحوظ بشكل أثرت قد عليها المسلطة لألحمال التربة تحمل قيمة إن التربة نوع تأثير إن المقيدة الحركة من والواضح والمقيدة الحرة للحركة بالنسبة الحركة لنوع بالنسبة اختالف يسبب المستطيل إلى المربع حالة من الصلد البالط شكل تأثيره. إن يقل الصلد البالط أسفل

التشوه. لقيم بالنسبة جدا كبيره قيم ويعطي اإلنحاء تاجتهادا %( لقيم28.57) مقداره1-Introduction

Airport engineering involves design and construction of a wide variety of facilities for the landing, and parking of air craft, maintenance and repair of aircraft, fuel storage, and handling of passengers, baggage, and freight. Thus, at a typical airport, there are terminal buildings and hangars, pavements for aircraft runways, taxiways, and aprons roads [Harding , 2004].2-Rigid Pavements

Rigid pavements are mostly found in major highways and airports. They also serve as heavy–duty industrial floor slabs, port and harbor yard pavements, and heavy–vehicle park or terminal pavements. Rigid highway pavements, like flexible pavements, are designed as all–weather, long–lasting structures to serve modern day high–speed traffic. They offer high quality riding surface for safe vehicular travel, and

function as structural layers to distribute vehicular wheel loads that induced stresses transmitted to the sub grade soil are of acceptable magnitudes [Fwa, & Wei, 2006]. The rigid pavement design curves are based on Westergaard [1927] analysis of the edge loading. The edge loading analysis has been modified to simulate a jointed edge condition [Al-Sahhaf, 1971]. These are made of Portland Cement–Concrete, usually placed on a suitable subbase course. Use of the design curves for rigid pavements requires the flexural strength of the concrete, the k value of the subbase, the gross weight of the design aircraft , and the equivalent number of annual departures of the design aircraft [Harding, 2004].

The (k) value is based on the material directly beneath the concrete pavement. A (k) value should be established for the sub grade and the corrected to account for the effects of the subbase. Rigid pavement for small aircraft (weighting 12.500 lb or less) should be at least 5 in thick. For air craft between 12.500 and 30.000 lb, the rigid pavement should be at least 6 in thick. All paved areas should be considered as critical areas Reinforcing of the pavement is desirable to control cracks. Its installation should follow the latest design and construction practices [Harding, 2004]. 3-Flexible Pavements :

These consist of a bituminous surface course, a base course of suitable material, and usually a granular subbase course. Design of flexible pavements is based on the results of sub grade soils tests. The FAA has developed a relationship between soil classes and thickness of surface course, base course, and subbase course required for various gross weights of aircraft, based on different conditions of drainage and frost action [Harding, 2004]. 4-Finite Element Solution

The philosophy of the finite element formulation (that pioneered by Clough and others) Zienkiewiez, [1964], is a numerical approximation method, mode to the exact mathematical differential equation that governing the problems. By it, concentrating on a number of selected values of the unknown function at specified mesh points. Any structures (in this study pavement slab), is divided into a series of quadrilateral elements. These elements are then jointed at nodal points only and continuity together with equilibrium equations are established at these points.4-1 2–D Finite Element Solutions The finite element method provides a modeling alternative that is well suited for applications involving systems with irregular geometry, unusual boundary conditions or nonhomogenous composition. The most widely used finite element computer codes are based on the slab mosel that idealizes concrete pavement slabs as classical thin plates and a multi-slab system under the action of multiple loads at any locations supported by a sub grade idealized either as Winkler foundation or semi infinite elastic solid foundation [Fwa, & Wei, 2006].4-2 3–D Finite Element Solutions

With the rapid development of computer technology in recent year , There has been marked improvements in the computational efficiency and capability of computing devices . The use of three–dimensional (3-D) finite element programs for pavement analysis is no larger a forbidden task. The general–purpose 3-D finite element code ABAQUS offers a powerful analytical tool not available previously for solving complex pavement problems [Fwa, & Wei, 2006].

4-3 Limitations of Finite Element Solutions

In performing element analysis, the design of finite element mesh as well as convergence verification and accuracy assessment required skills and professional, knowledge of the technical nature of the problem analyzed. The right solutions and sufficiently accurate answers are usually obtained through a trial–and error process. each computer run of finite element analysis,3-D finite element analysis effort , and the analysis itself is highly demanding computationally even with today's high–speed computer technology [Fwa, & Wei, 2006]. 5. Outline of Computer Program

The computer program was had the main title {Elasto-Plastic Analysis of Thick Plates Using an Incremental Finite Element Method (with modified Newton-Raphson iteration)}, which existed it by Alwash, [1989]. In this study, the same program is used after modified it at three principal changes, they are; (a) Enter plane stress (i.e. can be calculated the displacements in x-and y-directions. (b) The internal virtual work integration is done by layered approach with respect to z-direction. And (c) Using a nonlinear-elastic model in analysis behavior of concrete material at compression region instead of elasto-plastic analysis for metals material. After completed all changes, the program is became called {Nonlinear-Elastic Analysis of Thick Flat Plates by Using Finite Element Method} and for abbreviate will be namely ABAS-PROGRAM. The ABAS-PROGRAM is coded in Fortran-77 language and operated it by Fortran Power Station 4.0.

The ABAS-PROGRAM consists of one main routine program, as shown in flow chart(Appendix A), and add two new subroutines are; (1) Subroutine Program is divided the thickness of plate into chosen of layers and calculated the distance from mid-thick plate (Z=0.0) to center of each layer, and (2) Subroutine program is solved the stiffness matrix of foundation for two case, (a) Winker model, (linear, and nonlinear analysis). Also, within some old subroutine, was added or changed some parameters (like that, change the domain of geometry matrix, add parameters to calculate the axial displacement at x-and y-axes).

6. Formulation of Isoparametric Element and Assumed Displacement Field

A thick flat shell element will be considered in the analysis of slab pavement. Therefore, this element is a combination between the plane stress with the plate bending elements. This resulting element is flat and has five degrees of freedom per node i, neglecting the normal shell rotation ( ).The natural coordinate system is a local coordinate system which specifies location at any point within the element by a set of dimensionless numbers whose magnitudes never exceed unity Desai and Able, [1972], see Fig.(3.1c). The isoparametric elements are element for which the geometry and displacement formulations have the same order (i.e., the interpolation of the element coordinates and displacements use the same interpolation function, which are defined in a natural coordinate system). An 8-noded isoparametric (serendipity) element is used and at each node the degrees of freedom are, Cook, [1981]; (1) The transverse displacements ( ), (2) A rotation in x-direction ( ) {of the line that was originally normal to the mid-surface}, (3) A rotation in y-direction ( ), (4) Displacement in x-direction ( ), and (5) Displacement in y-direction ( ). Sign conventions for such degrees of freedom are shown in Fig. (3.1). the displacements field of such element could be given as follows Cook, [1981].

………… (3.1a)

………… (3.1b) (Quadratic displacement field)where;

: Transverse displacement.

:Displacement in x-direction in the flat shell element, Fig.(3.1b).

:Displacement in y-direction in the flat shell element, Fig.(3.1b).

: Displacement in x-direction in slab pavement body, Fig.(3.1b).

: Displacement in y-direction in slab pavement body, Fig.(3.1b).

: Average rotation about x-axis.

: Average rotation about y-axis.

The (Ni) are shape function of and (isoparametric coordination) and given in table (1).The main features of this element are Hinton, [1980] and Alwash, [1989]:

1. It is based on Mindlin’s assumptions ,which are :a. Displacements are small compared with plate thickness.b. The stress normal to the mid-surface of the plate ( ) is negligible.c. Normal to the mid-surface before deformation remain straight (but not necessarily normal to the mid-surface) after deformation. However, this

●●

●

●

●

●

●● ●

Node iPlate thickness (t) Node i

Z

X

Y

iwioyi u,

ioxi v,● ●

●

●

●

●

●

●

1

-1.0 1.0

1.0

-1.0

2

34

5

6

7

8

(a) (b) (c)

Figure (1) (a) An 8-node serendipity element. (b) The degrees of freedom at a typical node i for shell element. (c) Plan view showing isoparametric coordinates ζ-η an element of general shape.

uv

ζ

η

assumption is modified so that cross-sectional warping is permitted through the new proposed expressions for the transverse shear strains, with using a shear correction factor, as will be seen later in this chapter.

2. The element is capable of producing correctly the constant curvature state, even if the element is of non-rectangular shape.

3. Interelement compatibility is satisfied since ( ) and ( ) is independent of

( ).4. Quick convergence of the resulting displacements as the mesh is refined

progressively.As indicated in Ref. Sawezuk, [1963], 8-noded element is capable with high accuracy in the majority applications especially for thick plate as in this study.

Table(1) Shape functions for the 8-noded plane quadratic isoparamtic element.Node(i) Corresponding Shape Function(Ni)

1 -0.25(1-)(1-)(1++)2 -0.25(1+)(1-)(1-+)3 -0.25(1+)(1+)(1--)4 -0.25(1-)(1+)(1--)5 0.5(1-2)(1-)6 0.5(1-2)(1+)7 0.5(1-2)(1+)8 0.5(1-2)(1-)

7. Sensitivity Analysis Many factors such as soil subgrade properties , pavement layers properties,

design aircraft characteristics, traffic characteristics and flexural strength of concrete affect the determination of rigid airfield pavement thickness. each factor is selected individually with others remained constant. The first factor is the load of wheel on determining pavement thickness according to prior FAA method, tire pressure to be in common use as reported by Aerodrome design manual and FAA Advisory [Westergaard, 1927]. The purpose of the sensitivity studies was primarily to select the refinement of the discretization and the approximations within the elements for the 3-D rigid pavement problem. This process involved producing a number of finite element models with varying mesh fineness and element types, solving those models to obtain approximate solutions , and observing the convergence trends for key response parameters such as bending stress, and deflection. Where possible, these responses were compared to analytical solutions and experimental results. For the sensitivity studies to be valid, the finite element models generated must be compatible with Westergaard's assumptions, all sensitivity studies for this research were conducted for the general problem of an elastic plate resting on a bed of springs foundation considering interior or edge loading conditions. All finite element solutions were compared against a Westergaard theory solution obtained from WESTER [Westergaard, 1927].7.1 Pavement Thickness

It is well known that the slab thickness is the major contributor to deflection and stress, by using ABAS-PROGRAM the effect of slab thickness on deflection at slab center and bending stress are presented in Figure (1) & (2), and input data are:

Elastic modulus, E = 20800 MPa Poisson Ratio, v = 0.15 Total distributed Load, p = 2.3769 MPa Sub-grade Reaction value, K = 36.67 MPa Number of Element Mesh = 4 * 4

Figure (1) Effect of Slab Thickness on Bending Stress

Figure (2) Effect of Slab Thickness on Deflection

7.2 Length to Width Ration (L/W)By using ABAS-PROGRAM the effect of dimension ratio on deflection at

slab center and bending stress are presented in Figure (3) & (4), and input data are: Elastic modulus, E = 20800 MPa Poisson Ratio, v = 0.15 Total distributed Load, p = 2.3769 MPa Sub-grade Reaction value, K = 36.67 MPa Number of Element Mesh = 4 * 4

Figure (3) Effect of Dimension Ratio on Bending Stress

Figure (4) Effect of Dimension Ratio on Deflection

7.3 K-Value (Subgrade Reaction)By using ABAS-PROGRAM the effect of subgrade reaction value on

deflection at slab center and bending stress are presented in Figure (5) & (6), and input data are:

Elastic modulus, E = 20800 MPa Poisson Ratio, v = 0.15 Total distributed Load, p = 2.3769 MPa Slab Thickness, t = 0.2286 m Number of Element Mesh = 4 * 4

Figure (5) Effect of Sub-grade Reaction on Bending Stress

Figure (6) Effect of Sub-grade Reaction on Deflection8 .Conclusions

From the results shown in previous paper the following conclusions can be drawing:1. The nonlinear material elastic analysis by finite element method is stable for

thin to thick plate and given excellent results when comprised with KENSLAB-program.

2. The slab pavement thickness, a single slab model is present large values, but for fixed and pin case are converge amount them its for deflections and bending stresses value.

3. The type of soil foundation is effected widely on values of deflections and bending stresses for fixed and pin conditions.

4. The aircraft structural is important taken the mechanism moving state in slab pavement study.

9. Recommendations 1. Study the response of the rigid pavement slab-joint-foundation system for

aircraft structure with nonlinear elastic on nonlinear foundation support.2. Study the efficiency of transition the transverse shear stresses from slab

loading to adjacent slab through subgrade foundation.3. Within this model, study the motility slab pavement system due to temperature

effect and taken this stat with joint types.

10. ReferencesAlwash, N. H., "Elasto-Plastic and Limit Analysis of Thich Plate", M. Sc. Thesis,

Baghdad Unniversity, 1989.Al-Sahhaf, N. A., "Computer Aid of Rigid Airfield Pavement" USA Department of

Transportation, 1971.Cook, R. D., "Concept and Applications of Finite Element Analysis", John Wiley and

Sons Inc., 2nd ed., 1981. Desai, C. S., and Able, J. F., "Introduction to the Finite Element Method", New York:

Van No strand Reinhold Company, 1972.Fwa, T.F. & Wei, L., "Design of Rigid Pavements", 2006.Harding, R. "Standard Handbook for Civil Engineers", 2004.Hinton, E., and Owen, D. R., "Finite Element Software for Plates and Shells",

Department of Civil Engineering university Collage, Swansea, U. K., Ch. 2, pp. 270, 1984.

Westergaard , H. M., "Stresses in Concrete Runways of Airports", proceeding Highway Research Board, 1939.

Westergaard , H. M., "Theory of Concrete Pavement Design", proceeding Highway Research Board, 1927.

Zienkiewicz, O. C., and Cheung, Y. K., "The Finite Element Method for Analysis of Elastic Isotropic and Orthotropic Slabs", B.Sc. (Eng.), Ph. D., M. I. C. E., professor of civil Engineering, Wales University, 1964.

APPIDIX (A)

Start

Read data defining material properties, geometry, boundary conditions, and loading.

Set to zero arrays and vectors required for accumulation of data.

Read additional data required for Mindlin plate analysis

Evaluate the equivalent nodal forces for distributed loading.

A

By using layered approach to find the initial element stiffness matrices and assemble to find the initial stiffness matrix for thick flat shell element.

Apply a selected load from the type of the applied.

Solve the simultaneous equations system (equilibrium equation) by Gauss Elimination (in-core banded solution) to find the nodal displacements.

Compute the stresses tensor at each id-layer in each Gauss point within each element for nonlinear-elastic Mindlin plate.

Select the first load increment to be Pi+1 for first loop.

A

Seti=i+1

Update the element nonlinear-elastic stiffness matrix and assemble to find the updated stiffness matrix for thick flat element

Apply the selected load increment vector G

Lo

ad in

crem

ents

loop

Solve the simultaneous equations system (equilibrium equation) by Gauss Elimination (in-core bonded solution) to find resulting nodal displacements.

B

Extract the elements nodal displacements.

Elem

ent l

oop Set

e=e+1

P

Compute the Jacobian matrix for area integration.

Gau

ss p

oint

loop

Setg=g+1

E

SetL=L+1

Compute the increment strain.

Compute the stress tensor vector to be added to stress history.

Compute the total internal equivalent nodal forces.

Laye

r loo

p

B

All elements in layered is considered

No Yes C

All Gauss point is considered

Gau

ss p

oint

lo

op

No

All elements in layered is considered

Yes

Elem

ent l

oop

No

E

P

C

Evaluate the total nodal forces vector for the whole plate.

Yes

IsLoad incr. =Max-load

Stop

PrintNodal displacements.All stresses tensor at each mid-layer for each Gauss point within each element.

G

Yes

No

Load

incr

emen

ts

loop

Flow chart: The main routine program for nonlinear-elastic analysis of plates by using finite element method.